A fast direct solver for a fourth order finite difference scheme for Poisson’s equation on the unit disc in polar coordinates

Abstract

We present a fourth order finite difference scheme for solving Poisson's equation on the unit disc in polar coordinates. We use a half-point shift in the r direction to avoid approximating the solution at r = 0. We derive our scheme from analysis of the local truncation error of the standard second order finite difference scheme. The resulting linear system is solved very efficiently (with cost almost proportional to the number of unknowns) using a matrix decomposition algorithm with fast Fourier transforms.